Parallel torsion and $G_2,Spin(7)$ instantons

Instanton properties of the characteristic connection $\mathrm{\nabla }$ on an integrable $G_2$ manifold as well as instanton condition of the torsion connection $\mathrm{\nabla }$ on a $Spin(7)$ manifold are investigated. It is shown that for an integrable $G_2$ manifold with $\mathrm{\nabla }$-parallel Lee form the curvature of the characteristic connection is a $G_2$ instanton exactly when the torsion 3-form is $\mathrm{\nabla }$-parallel. It is observed that on a compact $Spin(7)$ manifold with $\mathrm{\nabla }$ closed torsion 3-form the torsion connection is a $Spin(7)$ instanton if and only if the torsion 3-form is parallel with respect to the torsion connection.